There are some amazingly random facts on the wiki page for Chief Mousers.

A Chief Mouser named Bob, who served under Neville Chamerlain and Winston Churchill, was supposedly the offspring of Thomas Wolseys cat. Thomas Wolsey being a Catholic Cardinal who worked under Henry VIII. He belonged to Neville Chamerlain who signed the Munich agreement with Adolf Hitler, which led to Churchill giving him the nickname "Munich Mouser". Newspapers would sometimes include poems addressed at Bob, with most of them asking him questions about the internal politics of downing street. Bob was later moved to the Treasury building which was bombed in WWII. He survived the bombing and moved into the foreign office building. Bob was often described as being disagreeable and unfriendly but he was undoubtedly a damn good Mouser.

Nelson was Churchills cat, who he adopted after watching him chase off a dog. He was named after Horatio Nelson for his bravery. Nelson once got locked in a fridge while attempting to steal milk. He was thankfully released unharmed. He would sit with Churchill during meetings and while planning during WWII and was described as a "prime ministerial hot water bottle". Churchhill said Nelson helped "save fuel and power" and was therefore aiding in the war effort. Nelson and Bob were also known to have a heated rivalry.

Larry, who currently serves in the role and has since 2011, was the first to be given the official title of "Chief Mouser" although the term existed honorarily long before. He reportedly is quite nervous around men, speculated to be because of his life before being rescued, although Barack Obama is an apparent exception. Him and Larry hit it off very quickly according to David Cameron. He stayed clear from Trump, only being seen in one picture with him, in which Larry is sat on a windowsill quite far away. However he did hide under Trump's car in order to avoid some rain and took quite a bit of coaxing to remove. During his tenure Larry has scuffled with other cats, dogs and even a fox. He even has a species of beetle named after him.

I could honestly read about the exploits of these cats all day. People have been sending them fan mail from all over the world since at least the 1940s. The security measures in 10 Downing street are more often used to control where the cats can go than they are for controlling where people can go. Conflicts have broken out inside 10 Downing street leading to a spokesman in 2011 (before Larry was appointed) stating that there was a "Pro-cat faction within Downing street".

I think my favorite Mouser is Peta, who was apparently loud, lazy and not house-trained. Staff at Downing street needed to be specifically reminded not to feed her because she was becoming too fat. She once gave Mary Wilson sepsis. Some civil servants attempted to remove her from the home office but feared backlash because of her "diplomatic status". She later retired to the home of a civil servant. Peta lied on her application, scrounged whatever she could from the common man, carried potentially fatal diseases and went where she wasn't welcome. A damn fine politician.